THE KURAMOTOSIVASHINKY EQUATION JOHN C. BAEZ STEVE HUNTSMAN AND CHEYNE WEIS The KuramotoSivashinsky equation

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THE KURAMOTO–SIVASHINKY EQUATION
JOHN C. BAEZ, STEVE HUNTSMAN, AND CHEYNE WEIS
The Kuramoto–Sivashinsky equation
ut=−uxx −uxxxx −uxu
applies to a real-valued function uof time t∈Rand space x∈R. This equation was introduced as a
simple 1-dimensional model of instabilities in flames, but it turned out to mathematically fascinating
in its own right [7]. One reason is that the Kuramoto–Sivashinsky equation is a simple model of
Galilean-invariant chaos with an arrow of time.
We say this equation is ‘Galilean invariant’ because the Galiei group, the usual group of sym-
metries in Newtonian mechanics, acts on the set of its solutions. When space is 1-dimensional,
this group is generated by translations in tand x, reflections in x, and Galilei boosts, which are
transformations to moving coordinate systems:
(t, x)7→ (t, x −tv).
Translations act in the obvious way. Spatial reflections act as follows: if u(t, x) is a solution, so is
−u(t, −x). Galilei boosts act in a more subtle way: if u(t, x) is a solution, so is u(t, x −tv) + v.
Figure 1 — A solution u(t, x) of the Kuramoto–Sivashinsky equation. The variable xranges over
the interval [0,32π] with its endpoints identified. Initial data are independent identically
distributed random variables, one at each grid point, uniformly distributed in [−1,1].
We say the Kuramoto–Sivashinsky equation is ‘chaotic’ because the distance between nearby
solutions, defined in a suitable way, can grow exponentially, making the long-term behavior of a
solution hard to predict in detail [4]. And finally, we say this equation has an ‘arrow of time’
Department of Mathematics, University of California, Riverside CA, 92521, USA
Centre for Quantum Technologies, National University of Singapore, 117543, Singapore
2500 Valley Drive, Alexandria, Virginia, 22302, USA
James Franck Institute and Department of Physics, University of Chicago, Chicago IL, 60637, USA
E-mail addresses:baez@math.ucr.edu, sch213@nyu.edu, cheyne42@uchicago.edu.
Date: April 30, 2021.
1
arXiv:2210.01711v1 [math.AP] 4 Oct 2022
摘要：

THEKURAMOTO{SIVASHINKYEQUATIONJOHNC.BAEZ,STEVEHUNTSMAN,ANDCHEYNEWEISTheKuramoto{Sivashinskyequationut=uxxuxxxxuxuappliestoareal-valuedfunctionuoftimet2Randspacex2R.Thisequationwasintroducedasasimple1-dimensionalmodelofinstabilitiesinames,butitturnedouttomathematicallyfascinatinginitsownright[7].Oner...

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